Geodesic
An integral estimate for the derivative of a rational function
Izvestiya. Mathematics, Tome 14 (1980) no. 2, pp. 257-273 
V. I. Danchenko. An integral estimate for the derivative of a rational function. Izvestiya. Mathematics, Tome 14 (1980) no. 2, pp. 257-273. http://geodesic.mathdoc.fr/item/IM2_1980_14_2_a2/
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     title = {An~integral estimate for the derivative of a~rational function},
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let there be given numbers $\alpha,q,\lambda,p$ and $n$, $0\alpha\infty$, $1\leqslant q\leqslant\infty$, $0\lambda\leqslant\infty$, $1$, $n=1,2,\dots$, and let $R(n,p)$ be the class of rational functions $\rho(z)$ of degree $\leqslant n$, analytic for $|z|\leqslant1$, with \begin{gather*} \|\rho\|_p=\biggl(\,\int_{|\zeta|=1}|\rho(\zeta)|^p\,|d\zeta|\biggr)^{1/p}\leqslant1\\ (\|\rho\|_\infty=\sup\{|\rho(z)|:|z|=1\}). \end{gather*} It is proved that, if $\alpha\geqslant1+p^{-1}-q^{-1}$, then $$ \sup\biggl\{\biggl[\,\int_0^1(1-r)^{\alpha\lambda-1}\biggl(\,\int_0^{2\pi}|\rho(r\cdot e^{i\varphi}|^q\,d\varphi\biggr)^{\lambda/q}\,dr\biggr]^{1/\lambda}:\rho\in R(n,p)\biggr\}\infty. $$ Bibliography: 6 titles.

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[2] Dolzhenko E. P., “O zavisimosti granichnykh svoistv analiticheskoi funktsii ot skorosti ee priblizheniya ratsionalnymi funktsiyami”, Matem. sb., 103(145):1(5) (1977), 131–142 | MR | Zbl

[3] Flett T. M., “The dual of an inequality of Hardy and Littlewood and some related inequalities”, J. Math. Anal. and Applic, 38:3 (1972), 746–765 | DOI | MR | Zbl

[4] Privalov I. I., Granichnye svoistva analiticheskikh funktsii, GITTL, M., L., 1950

[5] Uolsh Dzh. L., Interpolyatsiya i approksimatsiya ratsionalnymi funktsiyami v kompleksnoi oblasti, IL, M., 1961 | MR

[6] Gvaradze M. I., “O prostranstvakh $B(p,q,\lambda)$ analiticheskikh funktsii”, Soobsch. Akademii nauk Gruzinskoi SSR, 77:2 (1975), 273–276 | MR | Zbl